Contingent Debt and Investment

Roman Goncharenko∗

October 11, 2016

Preliminary Version

ABSTRACT

This paper studies the implication of contingent debt (CoCo) on banks’ optimal investment

policy. CoCo is found to mitigate the standard debt-overhang problem when used to finance

investment. On the other hand, having contingent debt outstanding prior to investment leads

to the delay of investment via two channels: increased debt-overhang owning to the fact that

contingent debt is riskier than standard debt and dilution cost associated with the conversion of

contingent debt. The paper argues that CoCo, as a new regulatory tool which is meant to increase

banks’ stability, can negatively affect banks’ incentives to invest.

∗The Vienna Graduate School of Finance

1 Introduction

Since 2009 more than 160 banks (primarily from the EU) made about 350 issuances of contingent

convertible bonds (CoCos) in the total amount exceeding $335bn1. The introduction of CoCo by

regulators (Basel Committe (2010)) aimed at improving banks’ stability. However, a potential effect

of CoCos on banks’ investment seemed not to be considered. By examining the effect of CoCo

financing on banks’ optimal investment policy, I show that contingent convertibles can negatively

affect banks’ incentives to invest. Given a special role of banks as fund providers, the decreased

incentives to invest may have potentially detrimental effect to the rest of the economy.

The financial crisis of 2007-2009 has demonstrated that banks’ capital requirements were in-

adequately low in order to insure banks’ reasonable loss absorption capacity. As a result of such

an experience banking regulators turned to significantly higher capital requirements introduced by

Basel Committe (2010). In order to lower the cost of meeting these capital requirements and to en-

sure a swiftly recapitalization of banks without regulatory intervention a new prudential instrument

was introduced - contingent debt (a.k.a. CoCo2). As an alternative source of regulatory capital,

CoCo was first introduced in Flannery (2002) and Flannery (2005), which was later modified by a

number of authors as well as practitioners (e.g. Squam Lake (2010)).

CoCo is a hybrid debt instrument issued by a bank that absorbs bank’s losses in accordance

with their contractual terms once the capital ratio of the issuing bank drops below a prespecified

level. The idea behind CoCo is simple: it is meant to provide bank’s capital in the state it is

mostly needed - when capital is depleting. The two main characteristics of any CoCo is its loss

absorption mechanism and trigger level. As far as the loss absorption mechanism is concerned

there are two most common kinds: equity conversion (EC) and principal write down. Under the

EC loss absorption mechanism, upon reaching a prespecified trigger event the face value of CoCo

converts into equity according to a prespecified conversion rate. Alternatively, under the PWD

loss absorption mechanism, CoCo face value is written down upon reaching a prespecified trigger

event. CoCo’s trigger is usually defined in term of the common equity tier 1 (CET1) ratio. The

trigger event is the violation of the prespecified trigger level. Depending on a trigger level different

regulatory treatment of CoCo is possible3.

The first issuances of CoCo took place in 2009. Danske Bank A/S issued perpetual PWD CoCos

with the face value of about $4.4bn followed by Lloyds Bank’s issuance of both EC and PWD CoCos

with various maturities and trigger levels in the total amount of about $17bn. As of the mid of

2016 159 banks (primarily from the EU) made 347 issuances of CoCo in the total amount of more

than $335bn4. Most of CoCo issued are additional T1 capital instruments. The higher growth of

CoCo market is projected since banks have to meet new capital requirements by 2018.

As of today in most of the jurisdictions that have seen CoCo issuance the latter is treated

1Bloomberg2CoCo stands for contingent convertible. Not all contingent capital instruments are convertibles, some are written

down. However, I use the term CoCo as a synonym of contingent debt.3CoCos can count as either additional T1 capital if it is perpetual with trigger level above 5.125 % or T2 capital.4Bloomberg

2

as debt for tax purposes, that is, the interest payments on CoCo are tax deductible. One of the

last countries within the EU that has adopted the debt treatment of CoCos for tax purposes was

Ireland in the mid of 2015. However, there is still one jurisdiction that does not treat CoCos as

debt, and has not seen any CoCos issuances, which is the US. The paper by Goncharenko and Rauf

(2016), analyzing the sample of CoCo issuance in the EU during 2011-2016, argues that the tax

deductibility of CoCo was an important factor in whether a given jurisdiction saw CoCo issuances.

To study the effect of CoCos on banks’ investment policy I use a dynamic contingent claim

model a la Leland (1994) augmented with investment opportunity set similarly to Heckemeyer and

de Mooij (2013) and Sundaresan, Wang, and Yang (2014). In my model bank operates in the

regulatory environment characterized by the possibility of having its charter withdrawn should it

experience severe undercapitalization. The loss of the charter then leads to a costly resolution

process. The bank’s investment opportunity set is characterized by a growth option. The timing of

the growth option exercises is optimally chosen by the bank. The bank’s initial (prior to exercising

growth option) optimal capital structure consists of senior debt and CoCo. To cover the cost

associated with the growth option exercise the bank issues optimal combination of CoCo and

subordinated debt. Debt and CoCo have tax-deductible interest payments. Therefore, the bank

optimally chooses financing and investment policy that trades off the benefits of tax shields and

the cost of regulatory closure.

The impact of CoCo on the investment policy can be disentangled into two effects. The first

effect comes from the use of CoCo as a means to raise funds for investment purpose. This effect is

found to have positive impact on the bank’s investment incentives by mitigating the standard debt-

overhang problem. The second effect comes from having CoCo outstanding prior to investment.

This effect has a negative impact on the bank’s investment incentives via increased debt overhang

and the dilution cost of equity associated with CoCo conversion.

Having some debt outstanding induces bank to delay profitable investment due to the standard

debt overhang problem. Without CoCo there are only two ways the bank can finance the investment

outlay: junior debt or equity. Investing with equity produces no tax shields but at the same time

does not directly increase the likelihood of regulatory closure. Investing with equity, hence, will

increase the market value of debt the most, and thus, is associated with the highest severity of

debt overhang problem. Financing the investment outlay with subordinated debt, on the other

hand, will produce tax shields and, at the same time, will positively affect the likelihood of failure.

Because junior debt increases the investment payoff via tax shields and at the same time dilutes

the market value of senior debt via increased likelihood of failure, it is normally associated with

less sever debt-overhang problem. CoCo is a hybrid of debt and equity in that it produces tax

shields and does not increase the likelihood of failure. Therefore, investing with CoCo helps to

mitigate the standard debt-overhang problem and results in earlier investment than under junior

debt financing.

On the other hand, having CoCo outstanding prior to investment has an opposite effect on the

timing of investment. There are two channels through which outstanding CoCo can affect the bank’s

3

optimal investment policy. First, as any type of debt CoCo creates the standard debt-overhang

problem: a payoff from investment is shared among equityholders, debtholders and CoCo holders.

In fact, given that CoCo has higher coupons per principal (i.e. riskier) than senior debt, the debt

overhang associated with CoCo is even severer than that coming from senior debt. Secondly, if the

bank has an EC CoCo outstanding prior to investment then investing with any debt has a negative

value on shareholders’ equity through the dilution cost associated with CoCo conversion. Issuing

subordinated debt, or any debt for that matter, will increase the likelihood of the conversion, and,

hence, the dilution cost associated with it. The strength of the effect depends on the conversion

rate, of course: for a PWD CoCo, for example, there is no dilution cost at all. The debt overhang

problem associated with a PWD CoCo is stronger than that of an EC CoCo because the latter is

less risky than the former. However, investment still takes place earlier when the bank has a PWD

CoCo outstanding thanks to the absence of the equity dilution cost. Therefore, this paper identifies

the superiority of the PWD loss absorption mechanism over the EC one in terms of the effect of

CoCo on investment policy.

The robustness of the effect of CoCo on bank’s investment policy is examined for different values

of the model parameters. I find that the results persist for realistic set of parameter constellation.

The model has also predictions regarding bank’s preference over CoCo’s loss absorption mech-

anism. I find that banks with more volatile assets opt for an EC CoCo, whereas their counterparts

with lower asset volatility optimally issue a PWD CoCo. Moreover, in the model, the amount

of CoCo issued negatively depends on bank’s asset volatility. Both facts have been empirically

confirmed by Goncharenko and Rauf (2016).

The rest of the paper is organized in the following manner. Next I review literature on CoCo.

Section 2 presents the general model. Next, Section 3 discusses the model’s assumptions. Section 4

presents the analysis of the simplified model. Finally, Section 5 examines the robustness of the

results to different parameter values.

Literature Review

The literature on CoCos, which is surveyed below, has blossomed in a rather short period of

time. Initially papers on CoCo focused primarily on two following issues. The first issue being the

effect of CoCos upon banks’ capital structure, though, without modeling CoCo as the part of the

optimal capital structure. In all of the papers analyzing the effect of CoCo on capital structure

either the choice of the entire capital structure is not modeled endogenously (e.g. Glasserman

and Nouri (2012a),Pennacchi (2010), Pennacchi, Vermaelen, and Wolff (2014), Chen, Glasserman,

Nouri, and Pelger (2015)) or the choice of CoCo (Albul, Jaffee, and Tchistyi (2013)). The second

issue being the pricing of CoCo (Sundaresan and Wang (2015) and Pennacchi and Tchistyi (2015)).

Recently, more attention has been payed to the effect of CoCo on risk incentives (Zeng (2014)).

Glasserman and Nouri (2012a) use Merton (1974)’s framework to study properties of CoCo

with accounting-based trigger. They show that whenever the trigger is reached the conversion of

4

CoCo into equity will occur by the amount just sufficient to meet the capital requirements. Yield

spreads are derived in a tractable way assuming that the trigger takes place prior bankruptcy. The

calibration shows that the fair yield for contingent capital, in their model, is sensitive to some of

the model’s inputs such as the size of the convertible tranche, to the volatility of the firm’s assets,

and to the recovery rates. They argue that this sensitivity as well as overall complexity of the

securities can become an obstacle to generating the investor demand.

Albul et al. (2013) (henceforth AJT), employ the model a la Leland (1994) in order to analyze

the implication of CoCo financing. The focus, as well as in the previous paper, on capital structure

decision. The trigger is assumed to be placed on firm’s asset value and the closed form solution

for the optimal capital structure including CoCo is derived. AJT show that introducing CoCo

into firms De Novo capital structure in an exogenous fashion, increases firms value by the tax

shield associated with CoCo, CoCo crowds out adjusted-to-tax-benefit equity one-to-one, and has

no effect either on straight debt or on default. AJT demonstrate that under the capital requirement

constraints CoCo reduces bankruptcy cost and may even increase firms value. Finally, AJT find

that if a leveraged firm replaces a fraction of straight debt with CoCos, the value of equity decreases,

firms value may increase , and, bankruptcy cost decreases.

AJT also show that CoCos can increase firm value and reduce the chance of costly bankruptcy or

bailout if properly implemented. However, shareholders of overleveraged or too-big-to-fail firms may

resist straight-debt-for-CoCo swaps due to the debt-overhang problem or the loss of the government

subsidy. The debt in the model is perpetual and, thus, any reduction in bankruptcy cost accrues

entirely to the initial debtholders.

Pennacchi (2010) uses a structural approach to model a bank that has short-term deposits,

common equity, and CoCo bonds. Banks asset follow a diffusion process with jumps.The short

term deposits are priced at par. The trigger is placed on a ratio of the asset value to the combined

value of CoCos and equity assuming that bankruptcy is costless. This helps him to get unique

equilibrium in stock and CoCo prices because the ratio is independent of the conversion of CoCos.

Pennacchi et al. (2014) adopt the framework by Pennacchi (2010) but propose a different design

of CoCo labeling it as COERC. COERC allows the bank’s original shareholders to buy back the

shares at the par value of the bond. They assume that the MM world holds and, hence, do not

model cost of financial distress, which obviously is important for banks.

Chen et al. (2015) use a contingent model to study the design of CoCo bonds and their incentive

effects in a structural model with endogenous default, debt rollover, and tail risk. The low tail risk

is modeled by letting banks asset value follow a diffusion process with downward jumps. They show

that once the firm issues CoCo the optimally chosen bankruptcy level can be at one of two levels:

(1) a lower level with a low default risk and (2) higher level at which default precedes conversion.

An increase in the firm’s total debt load can move the firm from the first regime to the second, a

phenomenon they call debt induced collapse because it is accompanied by a sharp drop in equity

value. By setting the trigger sufficiently high allows to avoid such a hazard. Since debt has finite

maturity in the model, in contrast to Albul et al. (2013), Chen et al. (2015) find that sometimes

5

the shareholders will want to issue CoCo bonds because the associated reduction in bankruptcy

cost will reduce the interest rate required on subsequent debt issues.

Himmelberg and Tsyplakov (2012) use a model of dynamic capital structure choice to analyze

how contractual terms of CoCo bonds affect future capital structure incentives. They pay special

attention to the loss absorption mechanism. In particular, they find that the principal write-

down leads create perverse incentives for banks to pursue higher leverage. On the other hand,

conversion to equity creates incentive for banks to pursue lower leverage in order to avoid dilution

costs. Finally, they show that banks may choose not to issue CoCos because large fraction of the

benefits is captured by existing bondholders. Himmelberg and Tsyplakov (2012)’s model is heavily

parametrized: many variables of interest, such as, total debt principal, coupons, default boundaries,

are set exogenously in order to calibrate certain ratios.

Hilscher and Raviv (2014) use Merton (1974)’s model to study the effect of contingent securities

on risk taking and default probability. They find that there is a reduction in the default probability

of a bank issuing a CoCo bond. They also find that appropriate choice of contingent capital

terms (such as conversion ratio) can virtually eliminate shareholders’ incentives to risk-shift (a

problem that is present when banks do not issue CoCos). Only CoCos with conversion-to-equity

loss absorption mechanism are analyzed. All securities issued by bank have the same finite maturity

(including deposits). The default boundary is specified exogenously.

Hilscher and Raviv (2014), as well as all the models surveyed so far, assume asset in place - there

is no investment decisions. These means, that most of the results with regard to debt-overhang

and risk-shifting can only be obtained in an implicit way in the style of Leland and Toft (1996),

that is to say, this analysis shows the mere potential of the incentive problem. Because of that, no

quantitative prediction on this ground can be made (it is important to note that all of the models

surveyed are though of as quantitative models).

Other papers deal with equilibrium price issues of CoCos. For example, an early CoCo proposal

(Flannery (2005)) specified that bondholders should receive shares worth the band’s par value of

day’s closing price. The idea was that such CoCo would be quite safe since investors would be

repaid in dollars or in an equivalent value of shares. Unfortunately, such a conversion mechanism

could give to short sellers an unusual opportunity. One then could bid down the share price

forcing conversion and increasing the number of shares outstanding. Alternatively, CoCos could

be converted into a number of shares implied by the trigger. Although, that fixes the previous

problem with conversion, it almost surely guarantees then that the CoCo holders will loose at the

conversion making CoCo very expensive (potential too expensive to have any demand at all).

The literature on CoCo with stock market triggers starts from Sundaresan and Wang (2015)

(SW henceforth). This paper concludes that if bank issues CoCo with a stock market trigger

then its stock price will either have multiple equilibria (if CoCo conversion terms are advantageous

to the CoCo holders) or none (if CoCo conversion terms are advantageous to the banks initial

shareholders). Later Glasserman and Nouri (2012b) (GN henceforth), using similar setting, confirm

the case of no equilibrium, but demonstrate that under conditions which deliver multiple equilibria

6

in SW, the unique equilibrium stock price emerges.

Paper by Pennacchi and Tchistyi (2015) (henceforth PT) contributes to this strand of literature

by, first, rationalizing the different predictions of SW and GN regarding CoCos that have conversion

terms advantageous to the CoCo holders. In particular, they show a counter example to the proof

by SW regarding multiplicity of equilibria and confirm the correctness of GNs unique equilibrium.

Secondly, PT show that if one assumes that CoCo is perpetual (papers by SW and GN assumed

finite maturity) and most of CoCos are perpetual then under realistic parameter constellation

even CoCos with conversion terms advantageous to the banks initial shareholders yield a unique

equilibrium of stock price. Moreover, PT identify the exact problem that led both SW and GN

to conclude nonexistence of equilibrium this is a lump sum principal payment at maturity that

destroys the existence of a unique equilibrium stock price when conversion terms are favorable to

initial equityholders.

McDonald (2013) suggests a dual price trigger. In particular, McDonald (2013) argues that a

CoCo should convert when two conditions are met: the firm’s stock price is at or below a trigger

value and the value of a financial institutions index is also at or below some trigger value. Such a

structure would potentially protect firms during a crisis, when all firms are doing badly, but at the

same time, in good times, it allows a bad bank to go bankrupt.

A paper by Bolton and Samama (2012) propose an alternative instrument to CoCo bond - a

capital access bonds (CAB). The basic idea is that a bank issuing a CAB effectively purchasing

rights to issue equity in crisis events at a pre-specified price form long term-investors. By doing

that, an issuing bank can ensure it will have sufficient regulatory capital when it needs it most

- in a crisis. Thus, this bond from the point of view of the issuer is an American option (can

be converted prior the maturity). If the bond is not converted it give investors a regular coupon

(interest plus put premium). If the bond is converted, then investors get a fixed number of newly

issued shares. Thus, this reverse convertible bond is actually a collateralized put option - because

the commitment to buy equity will be honored since the money are paid upfront.

To my knowledge, there have only been a few empirical papers analyzing CoCo bonds (Avdjiev,

Kartasheva, and Bogdanova (2013), Avdjiev, Bolton, Jiang, Kartasheva, and Bogdanova (2015)

and Berg and Kaserer (2015)). Most of the institutional details on CoCos in the first sections of

the paper have been borrowed from Avdjiev et al. (2013) and Avdjiev et al. (2015).

Avdjiev et al. (2013) is basically the first paper to present some basic empirical facts about

CoCo market. Avdjiev et al. (2015) builds on Avdjiev et al. (2013) by tremendously expanding

the sample of CoCo issues. They conduct the first comprehensive empirical study of the bank

CoCo issues market between 2009 and 2015. They find that it is the large banks with adequate

core capital that are among the earliest adopters of CoCo. Issuance of CoCos is documented to

an average of 8 basis point drop in the issuers’ CDS spread which indicates that CoCo issuance

reduces banks’ credit risk. The reduction of the CDS spread is shown to be increasing in the trigger

ratio. At the same time no significant impact on equity prices of CoCo issuers is detected. But the

trigger threshold is significantly positively (negatively) related to stock returns for principal write

7

down (conversion to equity) CoCos.

Berg and Kaserer (2015) use a simple option-pricing example to illustrate that the issuance of

CoCo bonds may increase equity holders’ incentive for risk-shifting and decrease incentives to raise

new equity during a crisis if there is a wealth transfer from CoCo investors to shareholders at the

conversion point. They then consider a relatively small sample of CoCo issuances and argue that

(1) almost all CoCo bonds designed in such a way that the above mentioned transfer is present and

(2) this contractual design is reflected in traded prices of CoCos (the sensitivity of the CoCo bond

total return with respect to a changes in implied volatility is approximately five times larger than

the corresponding sensitivity on the straight bond). They conclude, that CoCo bonds, given their

current design, may create perverse incentives for banks’ equity holders.

Zeng (2014) uses optimal contracting approach to study the effect of CoCo on banks efficiency

(ex post risk shifting) and the role of CoCo within banking regulation (the minimum capital require-

ments in particular). Paper argues that CoCo arises naturally in the equilibrium without regulation

as means to implement more superior from efficiency point of view contingent capital structure of

banks. Moreover, in the equilibrium with regulation (minimum capital requirements) CoCo emerges

naturally in order to efficiently implement optimal countercyclical capital requirements.

8

2 General Model

Bank holds a portfolio of risky assets (loans, securities, etc.) generating cash flows. The value

of such a portfolio is given by Vt at time t. The cash flow associated with the portfolio is denoted by

Yt. Under risk-neutral measure the dynamics of the cash flow follows geometric Brownian motion

with drift

dYt = µYtdt + σYtdWQ (1)

where µ denotes a drift of the process under risk-neutral measure, σ is the volatility of the cash

flow, WQ is the Brownian motion under risk-neutral measure Q. Let r denote the risk-free interest

rate. Then the after-tax value of the portfolio under risk-neutral measure is given by

V (Yt) = (1 − τ)EQ [∫∞

tYse

−rsds] = 1 − τr − µYt (2)

At t = 0, given the initial cash flow Y0, the bank can issue two types of debt: standard (senior)

debt and contingent-convertible (CoCo) debt. Let D0 (Y0) denote the face value of senior debt

and D0c (Y0) the face of CoCo. The interest payment on both types of debt is assumed to be tax-

deductible. In most of the jurisdictions that have seen large issuance of CoCo, mostly the EU, the

latter is treated as debt for tax purposes by the local fiscal authority. The bank issues senior debt

and CoCo trading off tax benefits to both types of debt and the cost of financial distress.

The bank has a growth option. The cost associated with the exercise of the growth option is

assumed to be a constant I > 0. When the growth option is exercised the bank’s assets value, V ,

is scaled up by a multiplicative factor to (1 + φ)V , where φ > 0 measure the profitability of the

growth option. For sufficiently large value of I the bank find it optimal to wait before exercising

the growth option. Thus, the bank has to optimally decide on when to invest and how to cover the

investment outlay I. Suppose that the bank invests at some random time T i when its cash flow

first time riches a certain threshold Y i, discussed below. The fixed cost I is financed by optimally

chosen combination of junior-debt, Dis (Y i), CoCo, Di

c (Y i), and equity.

The balance sheet of the bank prior to investment, that is for t ∈ [0, T i], is presented in Table 1.

Given the value of the bank’s assets, V (Yt) and the book value of the risky debt, D0 (Y0), and

CoCo, D0c (Y0), bank’s tangible (regulatory) equity, CET1, is defined as the difference between

bank’s total assets and total debt, CET1 = V (Yt) −D −Dc. Let, F 0 (Yt) denote the total value

of bank and E0 (Yt) denote the market value of bank’s equity, both are prior to investment. The

charter value of the bank, CV 0 (Yt) can then be defined as the difference between bank’s value

F 0 (Yt) and the value of the assets V (Yt).The balance sheet of the bank after investment has been made, that is for t ∈ (T i,∞), is

presented in Table 2. The bank’s asset value now is given by (1 + φ)V (Yt). Since when exercising

the growth option bank can issue subordinated debt and CoCo its regulatory equity is defined as the

difference between its asset and the total debt (including newly issued one), CET1 = (1 + φ)V (Yt)−D0 (Y0)−D0

c (Y0). Let, F i (Yt) denote the total value of bank and Ei (Yt) denote the market value

9

of bank’s equity, both are after investment. The charter value of the bank, CV i (Yt) can then be

defined as the difference between bank’s value F i (Yt) and the value of the assets (1 + φ)V (Yt).

Asset Liability

Assets: V (Yt) Debt: D0 (Y0)CoCo: D0

c (Y0)Tangible Equity (CET1): V (Yt) −D0 (Y0) −D0

c (Y0)Charter Value: CV 0 (Yt) Equity: E0 (Yt)

Table 1 The bank’s balance sheet prior to investment. D0 (Y0) and D0c (Y0) denote the face value

of the bank’s senior debt and CoCo, respectively. CV 0 (Yt) denotes the charter value of the bankwhich is define as the dereference the bank’s total value and its asset value. CET1 is the bank’sregulatory equity defined as the difference between its asset value and its total debt. Finally, E0 (Yt)denotes the market value of the bank computed as the difference of the bank’s total value and themarket value of its total debt. The market value of equity includes the value of regulatory equity.

Asset Liability

Assets: (1 + φ)V (Yt) Debt: D0 (Y0), Dis (Y i)

CoCo: D0c (Y0), Di

c (Y i)CET1: (1 − φ)V (Yt) −D0 (Y0) −D0

c (Y0) −Dis (Y i) −Di

c (Y i)Charter Value: CV i (Yt) Equity: Ei (Yt)

Table 2 The bank’s balance sheet after investment. D0 (Y0), Dis (Y i), D0

c (Y0), and Dis (Y i)

denote the face value of the bank’s senior debt, subordinated debt, CoCo issued prior to, and afterinvestment, respectively. CV i (Yt) denotes the charter value of the bank which is define as thedereference the bank’s total value and its asset value. CET1 is the bank’s regulatory equity definedas the difference between its asset value and its total debt. Finally, Ei (Yt) denotes the marketvalue of the bank computed as the difference of the bank’s total value and the market value of itstotal debt. The market value of equity includes the value of regulatory equity.

CoCo is characterized by a conversion trigger. Once this trigger is breached CoCo is either

converted into equity (given a conversion rate) or is written down, depending on its loss absorption

mechanism. The conversion trigger is typically placed on the CET1 ratio, which is defined as the

ratio of CET1 to the bank’s assets. This means, that when the CET1 ratio of the issuing bank falls

below qc, the trigger is breached. Because the bank’s assets value is the function of the cash flow,

it is possible to identify thresholds Y c0 and Y c

i that are associated with the breach of trigger prior

to and after investment, respectively. When the cash process reaches Y c0 first time from above the

conversion trigger of CoCo with the face value D0c (Y0) will be breached. Similarly, when the cash

process reaches Y ci first time from above the conversion trigger of CoCo with the face value Di

c (Y i)will be breached.

Mathematically, the thresholds Y c0 and Y c

i can be obtained in closed from equating the CET1

ratio to the conversion trigger level before investment

V (Y c0 ) −D0 (Y0) −D0

c (Y0)V (Y c

0 )= qc, (3)

10

after investment, assuming CoCo issued before investment did not convert,

(1 + φ)V (Y ci∣nc) −D

0 (Y0) −D0c (Y0) −Di

s (Y i) −Dic (Y i)

V (Y ci∣nc

)= qc (4)

and after investment, assuming CoCo issued before investment converted,

(1 + φ)V (Y ci∣c) −D

0 (Y0) −Dis (Y i) −Di

c (Y i)

V (Y ci∣c)

= qc (5)

Therefore, Y ci can take either value Y c

i∣c or Y ci∣nc depending on whether the CoCo issued prior to

investment converted or not, respectively. For later use, define index j ∈ {c, nc} which is equal to

nc if CoCo issued at date t = 0 does not convert prior to investment, and is equal to c, if it does so.

Bank operates in the regulatory environment facing a possibility of being closed by the charter

authority. The closure rule is based on the regulatory capital which is measured by the CET1

ratio. For example, in the US the FDIC categorizes a bank as undercapitalized if bank’s capital

drops below 2 % of asset value. In this case a bank’s chartering authority normally closes the bank

(unless there is a good reason to believe the bank can be recapitalized) Shibut, Critchfield, and

Bohn (2003). Obviously, the bank can be closed either before or after the growth option has been

exercised.

Once again, because the bank’s assets value is the function of the cash flow, it is possible to

identify thresholds Y 0,r and Y i,rj that are associated with the regulatory closure prior to and after

investment, respectively. (Note that Y 0,r does not have subscript j since by the time the bank

fails prior to investment the CoCo issued at time t=0 will have already converted). When the cash

process reaches Y 0,r for the first time from above prior to investment the bank will be closed. The

bank will also be close when its cash flow falls below Y i,rj for the first time after the growth option

has been exercised.

Mathematically, the thresholds Y 0,r and Y i,rj can be obtained in closed from equating the CET1

ratio to the closure trigger level q before investment

V (Y 0,r) −D0 (Y0)V (Y 0,r) = q, (6)

and after investment(1 + φ)V (Y i,r

j ) −D0 (Y0) −Dis,j (Y i)

V (Y i,rj )

= q. (7)

The reason there is no CoCo in equations 6 and 7 is because by the time the CET1 ratio reaches

q the bank will not hold any CoCo, since qc > q.In order to be able to write down the valuation formulas for debt, CoCo, and equity, which I

need for solving the model, I define the prices of some Arrow-Debreu securities. The first Arrow-

11

Debreu security pays zero while the cash flow Yt is strictly above a certain threshold Y ∗ and pays 1

when this threshold is reached for the first time. which pays 0 if the cash flow Yt is strictly above a

certain threshold Y ∗ and 1 when this threshold is reached. Let denote the prices of such a security

as A (Yt, Y ∗) The price of such a security then is denoted by A(Yt, Y ∗), which functional form can

be found in appendix A.

Next define two thresholds Y ∗ and Y∗ such that Y ∗ > Y∗. Consider an Arrow-Debreu security

that pays nothing if the cash flow process Yt stays below Y ∗ and pays 1 if the process reaches Y ∗

for the first time from below, conditional that it did not reach Y∗ before. Denote the prices of such

a security by A (Yt, Y ∗, Y∗). The functional form of the price can be found in appendix A.

Finally, consider an Arrow-Debreu security that pays nothing if the cash flow process Yt stays

above Y∗ and pays 1 if the process reaches Y∗ for the first time from above, conditional that it did

not reach Y ∗ before. Denote the prices of such a security by A (Yt, Y ∗, Y∗). The functional form of

the price can be found in appendix A.

Having defined the Arrow-Debreu prices of the securities I next present the valuation formulas

for debt, CoCo, and equity. I first present the valuation formulas after the growth option has been

exercised and then the valuation formulas prior to investment. The valuation formulas are needed

to set up the bank’s optimization problem.

2.1 Valuations after Investment

The after tax value of assets after the growth option has been exercised, that is for any t ≥ Tic,is given by

V i(Yt) = (1 − τ)EQt [∫∞

t(1 + φ)Yse−rsds] = (1 + φ) 1 − τ

r − µYt (8)

The senior debt has infinite maturity and pays a constant coupon C0. The payment of the

coupon is terminated in the event of bank closure, in which case the debt-holders receive the value

of bank’s assets at closure minus the cost of the resolution process. If the bank’s assets at closure

minus the resolution cost exceed the face value of senior debt, senior debtholders enjoy full recovery

and the rest is payed to junior-debtholders. The cost of the resolution is assumed to be proportional

to bank’s asset value αV i (Y i,r), where α ∈ [0,1]. Therefore, at any point in time the market value

of senior debt after the exercise of the growth option is given by

D0,i (Yt) =C0

r(1 −A (Yt, Y i,r

j )) +min{D0 (Y0) , (1 − α)V i (Y i,rj )}A (Yt, Y i,r

j ) (9)

Equation 9 stipulates that senior debtholders receive the value of perpetuity C0c /r assuming no

regulatory closure and in the case of regulatory closure they receive the value of bank’s assets at

closure minus the resolution cost or the recover the face value in full.

The CoCo debt is assumed to be perpetual. The vast majority of all outstanding CoCos are

perpetual Avdjiev et al. (2015). CoCo pays a constant coupon to a CoCo-holder up until the

12

conversion trigger has been breached. In the case of trigger breach the CoCo converts either into

equity or is written down. The conversion rate is such that for each unit of the principal the CoCo-

holder receives λ units of shares. Additionally, the total amount of bank’s shares is normalized

to V i (Y i). This normalization insures the model is homogeneous of degree one in Yt. Similar

approach of modeling conversion rate of CoCo is used in Chen et al. (2015). Therefore, the market

value of the CoCo issued after investment at any point in time is given by

Dic (Yt) =

Cic,j

r(1 −A (Yt, Y c

i,j)) +λDi

c(Y i)V i (Y i) + λDi

c(Y i) + λD0c(Y0)I{j=nc}

Eic (Y ci,j)A (Yt, Y c

i,j)

(10)

where j ∈ {nc, c} controlling for the fact whether CoCo issued at t = 0 converted (c) or did not (nc).

Equation 10 stipulates that assuming no conversion the CoCo holder enjoys the value of perpe-

tuityCi

c

r . Assuming conversion the CoCo-holder receives λDic(Y i) bank’s shares. The total amount

of shares outstanding prior to conversion is normalized to V i (Y i), therefore the CoCo-holder owns

a fractionλDi

c(Yi)

V i(Y i)+λDic(Y

i)λD0c(Y0)Ij=nc

of the bank after conversion. If CoCo issued at t = 0 does not

converts prior to investment it converts simultaneously with CoCo issued after investment. The

market value of the equity immediately after conversion is denoted by Eic (Y ci,j).

Similarly, the value of CoCo issued at t = 0, assuming it did not convert prior to investment, is

given by the following equation

D0,ic (Yt) =

C0c

r(1 −A (Yt, Y c

i,nc)) +λD0

c(Y0)V i (Y i) + λD0

c(Y0) + λDic(Y i)E

ic (Y c

i,nc)A (Yt, Y ci,nc)

(11)

The subordinated debt has infinite maturity and pays a constant coupon Cs. The payment of

the coupon is terminated in the event of bank closure, in which case the debt-holders receive the

value of bank’s assets at closure minus the cost of the resolution process and minus the face value

of senior debt provided this value is positive, otherwise they receive nothing. Therefore, at any

point in time the market value of subordinated debt is given by

Dis (Yt) =

Cs,j

r(1 −A (Yt, Y i,r

j )) + ((1 − α)V i (Y i,rj ) −D0 (Y0))

+

A (Yt, Y i,rj ) (12)

where, (x)+ ∶= max{x,0}Intuitively, equation 12 says that the holder of the junior debt receives the value of perpetuity,

Cs/r assuming no regulatory closure and assuming closure the junior debtholders receive what is

left after the senior debtholders payed. If senior debtholders do not recover the full face value after

resolution cost, then junior debtholders receive nothing upon closure.

13

The value of the equity after investment is given by

Eij (Yt) =⎛⎝V i (Yt) − (1 − τ)

C0 +Cs,j +C0c I{j=nc} +Cic,jr

⎞⎠

+⎛⎝(1 − τ)

C0c +Cic,jr

− λD0c(Y0)

V i (Y i) + λD0c(Y0) + λDi

c(Y i)Eic (Y c

i,nc) I{j=nc}

− λDic(Y i)

V i (Y i) + λDic(Y i) + λD0

c(Y0)Ij=ncEic (Y c

i,j))A (Yt, Y ci,j)

+⎛⎝(1 − τ)

C0 +Cic,jr

− V i (Y i,rj )

⎞⎠A (Yt, Y i,r

j ) .

(13)

The first term in equation 13 is perpetuity of after-tax levered equity value if there is neither

conversion nor closure. The second term takes into account conversion. When conversion happens

equity holders gain from discontinuing CoCo payments but at the same time the value of equity

is reduced by dilution (if λ > 0). The third term takes into account bank’s closure. When closure

happens shareholders forgo future cash flows, gaining from terminating debt payments at the same

time.

Finally, I define the payoff from exercising growth option which is equal to the combined value

of equity, newly issued CoCo, junior debt, which is given by the following equation

F ij (Yt) = Eij (Yt) +Dic∣j (Yt) +D

is∣j (Yt)

= V i (Yt) + τCicr

(1 −A (Yt, Y c)) + τCis,j

r(1 −A (Yt, Y i,r

j ))

− (1 − τ) C0

r(1 −A (Yt, Y i,r

j ))

− (1 − τ) C0c

r(1 −A (Yt, Y c

nc)) Ij=nc

− λD0c(Y0)

V i (Y i) + λD0c(Y0) + λDi

c(Y i)Eic (Y c

i,nc) I{j=nc}A (Yt, Y ci,j)

−min{αV i (Y i,rj ) +D0 (Y0) , V i (Y i,r

j )}A (Yt, Y i,rj )

(14)

The first term in equation 14 is the after-tax value of the bank’s assets. The second and third terms

are the values of tax shield associated with newly issued CoCo and junior debt. The fourth term is

the net value of the coupon payments associated with senior debt. The fifth term is the net value

of the coupon payments associated with initial CoCo. The sixth term is the cost of equity dilution

associated with CoCo issued at t = 0. Finally, the last term captures the loss of future cash flow

due to regulatory closure.

14

2.2 Valuations prior to investment

The value of senior debt issued at date prior to investment, that is for all t < Ti, is given by the

following equation

D0 (Yt) =C0

r(1 −A (Yt, Y i

nc, Yc

0 ) (A (Y c0 , Y

ic , Y

0,r) + A (Y c0 , Y

ic , Y

0,r)A (Y ic , Y

i,rc ))

−A (Yt, Y inc, Y

c0 )A (Y i

nc, Yi,rnc ))

+ (1 − α)V 0 (Y 0,r)A (Yt, Y inc, Y

c0 )A (Y c

0 , Yic , Y

0,r)+ (1 − α) (V i (Y 0,r

c )A (Yt, Y inc, Y

c0 ) A (Y c

0 , Yic , Y

0,r)A (Y ic , Y

i,rc )

+V i (Y 0,rnc ) A (Yt, Y i

nc, Yc

0 )A (Y inc, Y

i,rnc ))

(15)

The first term in equation 15 is the value of the perpetuity of debt payments assuming no closure

prior to or post investment regardless of whether the initial CoCo converted. The second term

is what debtholders receive if closure takes place prior to investment. Finally, the third term is

debtholders payoff if bank is closed after it has exercised its growth option accounting for the fact

that the initial CoCo could either convert or not prior to investment.

Next consider the value of CoCo issued at t = 0. This value is given by the following equation

D0c (Yt) =

C0c

r(1 −A (Yt, Y i

nc, Yc

0 ) − A (Yt, Y inc, Y

c0 )A (Y i

nc, Yci∣nc))

+ λD0c(Y0)

V i (Y i) + λD0c(Y0)

E0 (Y c0 )A (Yt, Y i

nc, Yc

0 )

+ λD0c(Y0)

V i (Y i) + λD0c(Y0) + λDi

c(Y i)E0 (Y c

0 ) A (Yt, Y inc, Y

c0 )A (Y i

nc, Yci∣nc)

(16)

Equation 16 stipulates that assuming no conversion either prior to or post investment the CoCo

holders of CoCo issued at t = 0 enjoy the perpetuity value associated with the security’s coupon

payments. Assuming that conversion takes place prior to investment the CoCo holders receive

λD0c(Y0)/ (V i (Y i) + λD0

c(Y0)) fraction of the equity. Assuming that conversion take place after

the investment has been made the CoCo holders receive λD0c(Y0)/ (V i (Y i) + λD0

c(Y0) + λDic(Y i))

fraction of the equity, which is smaller than the previous one since now equity is also diluted with

the conversion of the CoCo issued at t = T i.

15

Finally, I present the value of equity prior to investment which is given by the following equation

E0 (Yt) = (V 0 (Yt) − (1 − τ) C0 +C0

c

r)

+ ((1 − τ) C0c

r− λD0

c(Y0)V i (Y i) + λD0

c(Y0)E0 (Y c

0 ))A (Yt, Y inc, Y

c0 )

+ ((1 − τ) C0

r− V 0 (Y 0,r))A (Yt, Y i

nc, Yc

0 )A (Y c0 , Y

ic , Y

0,r)

+⎛⎝F i (Y i

nc) +⎛⎝(1 − τ)

C0j

r− V 0 (Y i

nc)⎞⎠

+((1 − τ) C0c

r− λD0

c(Y0)V i (Y i) + λD0

c(Y0)E0 (Y c

0 )) − I) A (Yt, Y inc, Y

c0 )A (Y i

nc, Yi,rnc )

+⎛⎝F i (Y i

c ) +⎛⎝(1 − τ)

C0j

r− V 0 (Y i

c )⎞⎠− I

⎞⎠A (Yt, Y i

nc, Yc

0 ) A (Y c0 , Y

ic , Y

0,r) .

(17)

The first term in equation 17 is the perpetuity of after-tax levered equity assuming no closure, no

investment, and no CoCo conversion. The second term takes into account CoCo conversion prior to

investment. In this case the shareholders’ equity is diluted by new shares issued upon conversion,

but at the same time the shareholders gain from terminating the coupon payments. The third

term takes closure prior to investment into account. By this time CoCo has already converted,

therefore, the shareholders gain from discontinuing senior debt payments but forgo the future cash

flow. Finally, the fourth and fifth terms take investment into account. The fourth term assumes

that CoCo does not convert prior to investment, whereas, the fifth term assumes the opposite.

Upon investment, the shareholders receive the newly maximized value of the bank, F i (Y ij ), incur

costs I, and loose the perpetuity of the after-tax levered equity prior to investment.

Finally, the value of the bank prior to investment, F 0 (Yt), is then given by the sum of the value

of senior debt, CoCo, and equity for all t ≤ Tij

F 0 (Yt) ∶= E0 (Yt) +D0 (Yt) +D0c (Yt) . (18)

2.3 Model’s Solution

The model is solved backwards. Given the initial capital structure which is characterized by

senior debt D0 (Y0) and initial CoCo D0c (Y0), the shareholders decide on how when to exercise the

growth option, that is, the optimal threshold Y ij , and how to finance it, that is, the shareholders

decide on optimal level of subordinated debt, Dis (Y i

j ), and new CoCo, Dic (Y i

j ).

Since the shareholders cannot commit to when they exercise the growth option, the optimal

time to invest is characterized by the smooth-pasting condition (see Hackbarth and Mauer (2011)

and Sundaresan et al. (2014)). The smooth-pasting condition equates the marginal value of equity

prior to investment evaluated at the optimal investment threshold to the marginal payoff from

16

investment evaluated at the same threshold. Mathematically,

∂

∂YtE0 (Yt) ∣Yt=Y i

nc= ∂

∂YtF inc (Yt) ∣Yt=Y i

nc

∂

∂YtE0 (Yt) ∣Yt=Y i

c= ∂

∂YtF ic (Yt) ∣Yt=Y i

c

(19)

Solving the system of two nonlinear equations with two unknowns in equation 19 produces the

optimal investment thresholds Y inc, when initial CoCo does not convert, and Y i

c , when it does.

Simultaneously with choosing when to exercise the growth option, the shareholders decide on

how the investment outlay I is going to be financed. Given Y ij , decide on optimal amount of junior

debt, Dis∣j (Y

ij ), and CoCo, Di

c∣j (Yij ), raised solving the following optimization problem

max(Di

s∣j(Yij ),D

ij(Y

ij ))≥0

F ij (Y ij )

s.t. Dis∣j (Y

ij ) +Di

j (Y ij ) ≤ I

Eij (Y ij ) ≥ 0

Y ij is given

(20)

The constraint Dic∣j (Y

ij )+Di

j (Y ij ) ≤ I is important because it is possible that the shareholders will

find it optimal to raise CoCo/subordinated debt with the face value larger than the fixed cost I.

For low value of σ the optimal bank’s leverage is high. The bank cannot lever up prior to exercising

the growth option in this model. Therefore, the bank will use the exercise of growth option as an

opportunity to lever up to the optimal level. Hence, this intrinsic to the model capital structure

effect will interact with the investment decision. In fact, for low values of σ the constrained will

likely bind.

Solving optimization problem in equation 20, delivers the vector of the optimal investment

thresholds and the optimal allocation of junior debt and CoCo Θ∗ ∶= ({Y ij ,D

is∣j (Y

ij ) ,Di

j (Y ij )}j∈{nc,c})

as a function of the initial capital structure characterized by vector Γ ∶= (D0 (Y0) ,D0c (Y0)).

Given Θ∗, the optimal vector of the initial capital structure, Γ∗ is obtained from maximizing

the ex-ante total value of the bank. Formally, the following optimization problem is solved

maxΓ≥0

F 0 (Y0)

s.t. E0 (Y0) ≥ 0

Y0 and Θ∗ (Γ) are given

(21)

Finally, it is assumed that at any point in time no cash outflow (coupon payment) should exceed

the cash inflow. That is, the shareholders cannot top up the difference between outflows and inflows

of cash.

17

3 Discussion of the Model Assumptions

In this section I discuss the major assumptions of the model. Some of the assumptions are

crucial for the results whereas others are not as important as it may seem at first glance.

3.1 Deposits

In the model the main part of the bank’s liability is the senior debt which is modeled as a

standard risky debt. In reality, of course, the main bulk of banks’ liabilities are deposits. A way

to incorporate deposits into the model is to follow Sundaresan and Wang (2016)’s approach. They

model deposits as risk-free debt. Specifically, assume that the total amount of deposits is D. Since

deposits are risk-free they pay the risk-free rate r. But deposits also generate non-interest profits

(e.g. service fees). Suppose that the instantaneous cash-flow generated by deposits is proportional

to its face value ηD. Then the net payment to deposits is (r − η)D. Therefore, bank issues

deposits not only to maximize the tax shields associated with it, but also because it generates some

additional cash flow.

In order to justify risk-free nature of deposits, Sundaresan and Wang (2016) assume that deposits

are insured. Specifically, the assume that the instantaneous insurance fee is a constant and given

by I0 and satisfies the following equality

(1 −A (Yt, Y r)) I0

r= A (Yt, Y r) (D − V (Y r)) (22)

where Y r is the threshold at which the bank is closed by the regulator, and A (Yt, Y r) is the

Arrow-Debreu price of regulatory closure. Equation 22 stipulates that under the fair insurance the

regulators gain from receiving insurance premium), the left-hand side of the equation (is equal to

the cost of paying deposits in case of the bank’s closure (the right-hand side of the equation).

Introducing deposits into the model either instead of senior debt or together with it, although,

will result in quantitative predictions, it will not change the main results. With deposits instead of

senior debt the bank will select optimally higher leverage since deposits generate additional cash

flow.

3.2 Regulatory Closure

In my model I do not permit endogenous default. Although, endogenous defaults are presumably

common among non-banking corporations, it hardly ever happens in banking industry. It is unusual

for a bank to file bankruptcy. On the other hand, it is very common for a bank to have its license

withdrawn by its regulator when the former find itself severely undercapitalized. Therefore, instead

of allowing for endogenous default I assume a regulatory closure rule. The rule is such that the

bank is shut down once its capital ratio drops bellow a certain threshold qc, which is consistent

with the regulatory practice, for example, in the US (Shibut et al. (2003)).

However, having regulatory closure instead of endogenous default requires an assumption that

18

shareholders cannot inject equity. The ability of shareholder to inject equity, on the other hand,

hinges on the assumption that the shareholders have ”long pockets”, and therefore, can easily top

up the difference between the bank’s outflows and inflows of cash, provided it is profitable. But

in reality economic agents either face budget constraints or some funds may simply not be readily

available. Thus, it is important to understand that the opposite assumption to the one I make,

that is, the assumption of shareholders’ ”long pockets”, is even less realistic. Either assumption is a

strong one, in reality the shareholders most likely can top up the difference between the outflows and

inflows to some extend. I could relax my assumption of no equity injection, and instead assume that

some injection is possible (bot not unlimited). In this case the result will change quantitatively but

not qualitatively. What is crucial for my results, is that the shareholders do not have an unlimited

ability to inject equity, and, hence, face some constraints on the total outflow of cash.

3.3 Dynamic Capital Structure

In the model the bank optimally selects its capital structure at date t = 0 and is not allowed to

readjust it prior to investment. In reality, of course, firms do not wait till they invest to lever up.

Therefore, a potentially interesting modification of the model is to allow the bank to re-adjust the

capital structure prior to exercising the growth option. However, as Hackbarth and Mauer (2011)

argue, for a wide range of the model’s parameter values it is optimal for the shareholders to issue

new debt when investing, not prior to it, since that helps to fully exploit the tax advantage of debt

without worsening the debt overhang problem. The increase in debt coupon payments caused by

earlier issuance of new debt reduces post-investment equity value, thereby making underinvestment

problem more severe. Moreover, issuing additional debt prior to investment does not fully exploit

the incremental tax shield value inherent in the growth option because the optimal coupon and,

hence, the tax shield associated with it, is smaller than if the firm could wait and time the debt

issue with the investment decision.

In their analysis Hackbarth and Mauer (2011) assume that asset volatility, σ, varies between

0.2 and 0.4, which is a standard range for corporates. Banks’ asset volatility is much lower that

that, ranging from 0.04 to 0.15. The lower volatility of assets implies higher leverage. This means,

that on the one hand, if the bank waits sufficiently long before the exercise of the growth option

the bank’s leverage will deviate substantially from its optimal level, and hence the bank might find

it optimal to readjust its capital structure earlier. On the other hand, because the bank has high

leverage it might not find it optimal to readjust its capital structure before investment since that

would lead to a large reduction in equity value brought about by the increase in coupon payments,

and, hence, worsened underinvestment problem.

3.4 CoCo

In the model I assume that the characteristics of CoCo, the loss absorption mechanism and the

trigger level, are not the choice variables of the bank. In practice, banks have full discretion on

both of the type of loss absorption mechanism and partially discretion on the trigger level.

19

As far as the regular is concerned either a PWD CoCo or an EC CoCo both receive the same

treatment. Still, empirically we see that CoCo with different loss absorption mechanisms are issued.

Therefore in the last section I endogenize the choice of the loss absorption mechanism and examine

how it depends on bank’s characteristics (e.g. asset volatility, etc).

The trigger level is not bounded from above but it is so from below. For example, a common

practice in the EU is that for CoCo to be qualified as T1 capital its trigger level should not be

lower 5.125% in terms of CET1 ratio. The idea is that the higher the trigger level the better its

loss absorption ability is.

Although, the question of the optimal design of CoCo (from banks’ perspective) is very inter-

esting one, this is beyond the scope of this paper, which does not aim at rationalizing the bank’s

choice the trigger level. The aim of this paper is to analyze the effect of CoCos (as a new type

of security) on banks’ optimal (joint) financing and investment policy. Bringing the choice over

the CoCos’ characteristics will only make the analysis too complex without providing additional

benefits.

20

4 Analysis

In the model from section 2 the bank issues CoCo prior to investment and when investing.

Having some debt outstanding induces bank to delay profitable investment due to the standard debt

overhang problem. Without CoCo there are only two ways the bank can finance the investment

outlay: junior debt or equity. Investing with equity produces no tax shields but at the same time

does not directly increase the likelihood of regulatory closure. Investing with equity, hence, will

increase the market value of debt the most, and thus, is associated with the highest severity of

debt overhang problem. Financing the investment outlay with subordinated debt, on the other

hand, will produce tax shields and, at the same time, will positively affect the likelihood of failure.

Because junior debt increases the investment payoff via tax shields and at the same time dilutes

the market value of senior debt via increased likelihood of failure, it is normally associated with less

sever debt-overhang problem. CoCo is a hybrid of debt and equity in that it produces tax shields

and does not increase the likelihood of failure. Therefore, one could expect that investing with

CoCo should help to mitigate the standard debt-overhang problem and result in earlier investment.

On the other hand, having some CoCo outstanding could have the opposite effect on the timing

of investment. There are two channels through which outstanding CoCo can affect the bank’s

optimal investment policy. First, as any type of debt CoCo creates the standard debt-overhang

problem. In fact, given that CoCo has higher coupons per principal (i.e. riskier) than senior

debt, the debt overhang associated with CoCo may well be more severe. Secondly, if the bank has

some EC CoCo outstanding prior to investment then investing with subordinated has a negative

value on shareholders’ equity through the dilution cost associated with CoCo conversion. Issuing

subordinated debt, or any debt for that matter, will increase the likelihood of the conversion, and,

hence, the dilution cost associated with it. The strength of the effect depends on the conversion

rate: for a PWD CoCo, for example, there is no dilution cost at all.

The model from section 2 allows me to study the two effects of CoCos on tha bank’s optimal

investment policy. To simplify the analysis, I study the two effects separately. First, I assume that

initially bank issues no CoCo, that is using the notation from the previous section, D0c (Y0) = 0, and

finances the investment outlay either by CoCo or by subordinated debt. This allows me to isolate

the effect of investing with CoCo from the effect of having some CoCo outstanding. Moreover, I it

enables me to contrast the debt-overhang implications of CoCo and junior debt.

To study the implications of having CoCo outstanding on the optimal investment policy, I

assume that the investment outlay can only be financed by subordinated debt, that is, Dic (Y i) = 0,

but initially the bank issues both senior debt and CoCo. This approach allows me to isolate the

effect of having CoCo outstanding on the optimal investment policy without being distorted by the

effect of CoCo issued to financing the investment outlay.

The model from section 2 cannot be solved in closed from and, thus, the numerical solution is

obtained. To solve the model numerically I assign value to the model parameters discussed in the

next subsection.

21

4.1 Exogenous Parameters

In order to solve the model numerically it needs to be parametrized. The model has in total 9

parameters which are summarized by the vector (α,µ, σ, λ, τ, r, q, qc, Y0). The risk of debt, CoCo,

and equity is connected to the risk of assets via parameter σ - the volatility of the assets. Using

Moodys KMV estimates of firm’s asset volatility across many industries, Sundaresan and Wang,

2016, find that the mean volatility of bank’s asset during the period of 2001-2012 was about 8%.

Therefore, at the baseline value I set σ to 0.08 and then examine the sensitivity of optimal capital

structure to the volatility level.

The cost of resolution is basically a special type of the bankruptcy cost. The estimation of

the bankruptcy cost is a challenge in general. One of the first and highly cited references on the

estimation bankruptcy costs is Altman (1984), who finds that in a sample of 19 industrial firms

bankruptcy cost are about 20% of a firm’s value. Relatively recently, Bris, Welch, and Zhu (2006)

find that the cost of bankruptcy range from 0% to 20% within a comprehensive sample of small

and large corporate bankruptcies in Arizona and New York from 1995 to 2001.

In general, one could expect the cost of bank’s bankruptcy be larger than the one of an industrial

firm. Occasionally, when industrial firms go bankrupt they are still continuing their manufacturing

process without big changes. In contrary, a bankruptcy process for a financial firm often implies

a complete withdrawal from any business operations. This is mostly due to the stigma of failure:

a financial firm that is thought to experience some problems will face challenges with its creditors

and can become a subject of a bank run.

One of the first papers to reexamine the question of bankrupts cost in banking industry is James

(1991). Using the sample of the FDIC-regulated commercial banks failed during 1982-1988, James

(1991) finds that the cost of bank’s bankruptcy is about 30% of failed bank’s assets. Recently,

Flannery (2011) estimates bank’s bankruptcy cost to be about 26.6% of asset value. Given the

estimates of the bankruptcy cost, at the baseline value I set α to 0.25 and then examine the

sensitivity of the optimal capital structure to the cost.

The marginal tax rate is a parameter of crucial importance in capital structure, since one of

the main benefits of debt is the tax-deductibility of interest on it. Corporate tax rate differs among

jurisdiction. The federal tax rates on corporate taxable income vary from 15% to 39% in the US.

In the UK the corporate tax rate is a flat rate of 20% and Switzerland it ranges between 18% and

21%.

The US department of Treasury (2007) reports that on average the effective marginal tax rate

on investment in business is 25.5%, though, varying across industries. Graham (2000) estimates

the effective corporate tax rate for non-financial US firms to be around 10%. Heckemeyer and

de Mooij (2013) estimates the effective corporate tax rate for US commercial banks at about 30%.

Therefore, at the baseline value I set τ to 0.25 and then examine the sensitivity of the optimal

capital structure to this rate.

One of the main functions of the banking regulators is to monitor bank’s capital. Violating the

minimum capital requirement constraints usually leads to different sort of penalties imposed on

22

banks by their regulators. At some point bank’s capital can be so small that it is no longer viable

for the regulator to let bank continue its operations. Then bank’s charter authority may close the

bank. Particularly in the US, the FDIC categorizes bank as critically undercapitalized if bank’s

tangible equity-to-asset ratio hits 2 % level. In that case bank’s charter authority is likely to close

the bank appointing the FDIC as a receiver Shibut et al. (2003).

The CoCo security has two main characteristics: the trigger threshold qc and the conversion

rate λ. In the benchmark model I set the trigger threshold qc = 0.05125 which corresponds to

high-conversion CoCo. It is interesting to focus on high conversion CoCo since it has higher loss

absorption capacity. The conversion rate λ takes values in {0,2}, where 2 is for a EC CoCo and 0

is for a PWD CoCo. This allows me to study the effect of the loss absorption mechanism on the

optimal investment decision.

The risk-free rate, r, is set to 5% and the risk-neutral drift of the cash flow, µ, is set to 2%. The

initial value for the cash flow process, Y0, is calibrated such, that the asset value at t = 0, V (Y0) is

equal 1.

I set the investment outlay I = 0.5 and the profitability of the option growth φ = 0.65. This

implies that the NPV of the project at t = 0 is φ(1−τ)(r−µ)Y0 − I = 0.15 > 0. The growth option is

parametrized such that if the bank issues nether senior debt nor CoCo it will invest at t = 0. This

implies that the only reason the bank delays its investment is the debt overhang.

Table 3 presents the benchmark parameter values.

parameter value

Resolution cost α 0.25Risk-neutral drift µ 0.02Asset volatility σ 0.08Conversion rate λ {0,2}Marginal corporate tax rate τ 0.25Risk-free rate r 0.05Closure threshold q 0.02Trigger threshold qc 0.05125Initial asset value V (Y0) 1.00Cost of investment I 0.50Size of growth option φ 0.65

Table 3 The benchmark parameter values.

4.2 Investing with CoCo vs Junior Debt

Table 4 summarizes bank’s optimal financing and investment policy, and its implication on

bank’s characteristics, under four scenarios: when exercising its growth option bank issues (i)

junior debt, (ii) a PWD CoCo (λ = 0), (iii) a an EC CoCo (λ = 2), and (iv) equity. The coco-issuing

bank invests earlier than the junior-debt-issuing bank. This difference is economically significant.

If the bank raises funds by issuing a PWD CoCo, an EC CoCo, a junior debt, or equity, then

23

the investment is expected to take place in about four years, two years, eight years, and 17 years,

respectively. The probability of investment conditional on not being shut down prior to investing,

Πij (Y0), is higher for a CoCo financing (it the highest for a EC CoCo-issuing bank) and is the

lowest for equity financing. Although a CoCo-issuing bank tends to have higher initial leverage, its

conditional probability of investment is still higher due to a lower investment threshold.

Variable junior debt PWD EC Equity

D0j (Y0) 0.6418 0.6117 0.6680 0.6441

Dij (Y i

j ) 0.5000 0.5000 0.5000 0.0000

Y ij 0.0479 0.0437 0.0421 0.0579

Fj (Y0) 1.4143 1.4240 1.4296 1.3460RCj (Y0) 0.0099 0.0030 0.0041 0.0072CSD0

j0.04% 0.02% 0.03% 0.05%

CSDij

0.06% 0.22% 0.20% NaN

Πfj (Y0) 0.1273 0.0366 0.0425 0.0971

Πij (Y0) 0.9308 0.9671 0.9646 0.9040

T i 8.2490 4.1368 2.1641 17.4594

Table 4 compares bank’s optimal financing and investment policy, and its implication on bank’scharacteristics, under four scenarios: when exercising its growth option bank issues (i) junior debt,(ii) a PWD CoCo (λ = 0), (iii) a an EC CoCo (λ = 2), and (iv) equity. RCj (Y0) denotes thevalue of the resolution cost. TB0

j (Y0) and TBij (Y0) denote the value of tax shields from senior

and CoCo/subordinated debt, respectively. CSD0j

and CSDij

denote the credit spread in percentage

points for senior and CoCo/subordinated debt, respectively. Πfj (Y0) denotes the unconditional

probability of failure (the sum of the probability of failure prior to and after investment). Πij (Y0)

denotes the conditional probability of investment; the conditional probability of failure is then givenby 1 −Πi

j (Y0). The model is parametrized with benchmark parameter values from Table 3.

Under both CoCo and subordinated-debt financing the bank raises the investment outlay, I,

solely by debt. In order to understand why the growth option is exercised earlier under CoCo

financing than under junior-debt financing one needs to examine the smooth pasting condition

given in equation 19. At the optimal investment threshold the marginal value of equity is exactly

equal to the marginal value of the payoff ∂∂Yt

E0j (Yt) ∣Yt=Y i

j= ∂∂Yt

F ij (Yt) ∣Yt=Y ij. If the bank under

subordinated debt financing scenario is forced to exercise the growth option earlier - at the same

time that the CoCo-issuing banks does - then the marginal value of equity of the bank is smaller

than the marginal value of payoff, ∂∂Yt

E0s (Yt) ∣Yt=Y i

c< ∂∂Yt

F is (Yt) ∣Yt=Y ic, implying a convex kink at

the investment boundary. It can be shown numerically that the marginal value of equity under both

scenarios are roughly the same. Therefore, to understand why it is optimal for a subordinated-debt

issuing bank to invest later one needs to examine the the marginal value of the payoff closer.

24

The partial derivative of the payoff for a CoCo issuing bank is given by

∂

∂YtF ic (Yt) ∣Yt=Y ∗ ∶= (1 + ψ)(1 − τ)(r − µ) − τβ2

Cicr

A (Y ∗, Y c)Y ∗

+ (1 − τ)β2C0c

r

A (Y ∗, Y i,rc )

Y ∗− β2V

i (Y i,rc )

A (Y ∗, Y i,rc )

Y ∗

(23)

and for a junior-debt issuing bank it is given by

∂

∂YtF is (Yt) ∣Yt=Y ∗ ∶= (1 + ψ)(1 − τ)(r − µ) − τβ2

Cisr

A (Y ∗, Y i,rs )

Y ∗+ (1 − τ)β2

C0s

r

A (Y ∗, Y i,rs )

Y ∗

− β2min{αV i (Y i,rs ) +D0

s (Y0) , V i (Y i,rs )}

A (Y ∗, Y i,rs )

Y ∗.

(24)

The first term is the direct marginal benefit from waiting - as you wait assets grow bigger. The

second term is the marginal value of tax shield associated with new debt. Postponing investment

increases the value of this tax shield by decreasing the likelihood of loosing it. The third term is

the marginal value of the net payment to the senior debtholders. This term is increasing in Y ∗

due to convexity of the Arrow-Debreu price, thus, inducing to invest earlier. The last term is the

value of the marginal benefit from reducing the loss in closure. This term is decreasing in Y ∗ due

to convexity of the Arrow-Debreu price, thus, inducing later investment.

The first two effects happen to have roughly similar magnitude for both scenarios. The third ef-

fect is stronger for junior-debt financing because Y i,rs > Y i,r

c , and hence, A (Y ∗, Y i,rs ) > A (Y ∗, Y i,r

c ),

since CoCo does not contribute to the likelihood of closure directly. Thus, the effect from the third

term - the reduction in the value of the net payment to the senior shareholders - actually incen-

tivizes a junior-debt issuing bank to invest earlier. However, the opposite effect that comes from

the fourth term - the reduction in the value of the loss in closure - has a large magnitude offsetting

the previous effect and inducing later investment. The reason the marginal benefit from reduction

in the value of loss in closure is larger under junior-debt financing is again because the likelihood

of failure is quite larger comparing to that under CoCo financing.

CoCo is a hybrid of debt and equity. But it is a kind of a hybrid that takes the best from the

two types of securities. Similar to equity it does not contribute to the likelihood of bankruptcy.

Similar to debt it generates the tax shield. As a result, investment financed by CoCo suffer less

from debt-overhang problem.

The ex-ante value of the bank, Fj (Y0), is higher under CoCo financing scenario and the lowest

under equity financing. Under CoCo financing the value of tax-shields is higher and the value of

resolution cost is lower rationalizing a high ex-ante total value.

The total probability of bank’s failure, Πfj (Y0), is given by the sum of the probability of bank’s

failure prior to investing and the probability of bank’s failure after investing. The probability of

bank’s failure is the highest under subordinate-debt financing; it is lower under equity financing.

The probability is even lower for CoCo financing with the lowest value for a PWD-CoCo financing.

25

The credit spread associated with the senior debt under both subordinated-debt and equity

financing than under CoCo financing. When the growth option is financed by equity the credit

spread of senior debt are the highest partially because the initial leverage is slightly higher than

under subordinated debt, but mostly because the probability of failure prior to investing, 1−Πij (Y0),

is the highest. The lower senior-debt credit spreads under CoCo financing are explained by very

small probabilities of failure prior to investing despite the fact that initial leverage is somewhat

higher.

When investing with a PWD CoCo the bank issues less senior debt in order to be able to make

the coupon payments (recall that no equity injections are allowed in the model). Since a PWD

CoCo has higher coupon per face value than an EC CoCo, the bank optimally lowers the amount of

senior debt and, hence, the total coupon payment associated with it. In fact, given the benchmark

parameter values it is optimal for bank to opt for an EC CoCo.

4.3 CoCo Outstanding

In this subsection I present the results of the analysis of CoCo outstanding on the bank’s optimal

capital structure. I solve the model from section 2 with a restriction that the bank does not issue

CoCo when financing the investment outlay I, that is, Dic (Y i

j ) = 0. The model’s parameters assume

the values from Table 3.

Table 5 summarizes bank’s optimal financing and investment policy, and its implication on

bank’s characteristics, under three scenarios: (i) the benchmark scenarios under which the bank

does not issue CoCo at t = 0, (ii) the bank issues a PWD CoCo, and (iii) the bank issued an EC

CoCo.

As in the analysis from the previous subsection, the bank that has no CoCo outstanding is

expected to make the investment within about 8 years. In contrary, the bank that issues CoCo at

time t = 0 invests later, despite the fact that it actually has less senior debt. Although, a CoCo

issuing bank issues less senior debt, its total leverage is higher than that of non-issuing CoCo one.

Given that CoCo is riskier than senior debt - its coupon per face value is larger than that of senior

debt - it creates even more sever debt-overhang problem than senior debt.

Conditional on CoCo converting prior to investment a CoCo-issuing bank invests earlier than

non-issuing CoCo one. This can be seen from the values of the optimal investment thresholds Y ic

of the CoCo issuing bank: 0.0466 for a PWD CoCo and 0.0473 for an EC CoCo versus 0.0479

for the bank that does not have CoCo outstanding. This is because a CoCo-issuing bank does

not issues as much senior debt as the bank that does not issue CoCo. However, conditional on

CoCo not converting prior to investment, a CoCo-issuing bank invests much later, such that the

unconditional expected time before investment, T i, is larger when there is CoCo outstanding.

The bank issuing a PWD CoCo is expected to invest earlier than that issuing an EC CoCo:

about 10 years for the PWD CoCo issuing bank and about 13 years for the EC CoCo issuing

bank. This is because an EC CoCo is associated with the dilution cost of equity due to the

CoCo’s conversion. The larger the conversion rate λ, the higher the dilution cost. The dilution

26

cost, assuming that CoCo does not convert prior to investment, lowers the payoff from investment

resulting in the delay of the latter.

Variable No CoCo PWD EC

D0 (Y0) 0.6418 0.5958 0.6033D0c (Y0) 0.0000 0.1473 0.1684

Dis (Y i

j ) 0.5000 0.5000 0.5000

Y ij 0.0479 0.0486/0.0473 0.0521/0.0475

F (Y0) 1.4143 1.4483 1.4458RCj (Y0) 0.0099 0.0066 0.0061CSD0 0.04% 0.03% 0.03%CSD0

cNA 1.01% 0.79%

CSDis

0.06% 0.02%/0.04% 0.02%/0.04%

Πfj (Y0) 0.1273 0.0951 0.0938

Πij (Y0) 0.9308 0.9550 0.9434

Tj 8.2490 9.1704 11.7656

Table 5 compares bank’s optimal financing and investment policy, and its implication on bank’scharacteristics, under three scenarios: (i) the benchmark scenarios under which the bank doesnot issue CoCo at t = 0, (ii) the bank issues a PWD CoCo, and (iii) the bank issued an ECCoCo. RCj (Y0) denotes the value of the resolution cost. CSD0

j, CSD0

c, and CSDi

sdenote the credit

spread in percentage points for senior, CoCo and subordinated debt, respectively. Πfj (Y0) denotes

the unconditional probability of failure (the sum of the probability of failure prior to and afterinvestment). Πi

j (Y0) denotes the conditional probability of investment; the conditional probability

of failure is then given by 1−Πij (Y0). The model is parametrized with benchmark parameter values

from Table 3.

To better understand the implications of CoCo outstanding on the bank’s optimal investment

policy it is useful to examine the payoff from investment from equation 14. I rewrite this equation

under assumption that bank does not issue CoCo when exercising the growth option. For the bank

that issues CoCo initially, this payoff is given by the following equation

F ij (Yt) = V i (Yt) + τCis,j

r(1 −A (Yt, Y i,r

j ))

− (1 − τ) C0

r(1 −A (Yt, Y i,r

j ))

− (1 − τ) C0c

r(1 −A (Yt, Y c

nc)) I{j=nc}

− λD0c(Y0)

V i (Y inc) + λD0

c(Y0)Eic (Y c

i,nc) I{j=nc}A (Yt, Y ci,j)

−min{αV i (Y i,rj ) +D0 (Y0) , V i (Y i,r

j )}A (Yt, Y i,rj )

(25)

where, recall, index j ∈ {c, nc}, indicates whether the CoCo issued at t = 0 has converted prior to

investment, j = c, or has not, j = nc. The payoff from the investment of the bank that issues no

27

CoCo at date t = 0, on the other hand, is given by the following equation

F i (Yt) = V i (Yt) + τCisr

(1 −A (Yt, Y i,r))

− (1 − τ) C0

r(1 −A (Yt, Y i,r))

−min{αV i (Y i,r) +D0 (Y0) , V i (Y i,r)}A (Yt, Y i,r)

(26)

The payoff from investment of the bank with CoCo outstanding (equation 25) has two more

terms than the corresponding payoff of the bank that has no CoCo outstanding (equation 26).

These terms are the value of the net payments to CoCo holders - the fourth term in equation 25 -

and the cost of equity dilution - the fifth term in equation 25. Both of these terms have negative

effect on the payoff resulting in larger benefits from waiting before making investment. Even when

the conversion rate, λ, is zero, that is, when the bank has a PWD CoCo outstanding, the value of

the net payments to CoCo holders reduces the payoff substantially to induce later investment.

Table 5 shows that, although, the bank that has CoCo outstanding is expected to invest later

than the one that has no CoCo outstanding, the probability of investing is actually larger for the

former one. This has to do with the fact that the bank with CoCo outstanding optimally issues

less senior debt and, thus, faces lower likelihood of regulatory closure prior to investment.

The total probability of failure is the highest for the bank that issue no CoCo at date t = 0

since this is the bank with the highest level of senior debt. The PWD CoCo issuing bank has the

lowest total probability of failure thanks to relatively low level of senior debt issued. As a result

the ex-ante value of the bank is higher if it issues CoCo at date t = 0. This means that a value

maximizing bank optimally issues CoCo at date t = 0 despite the fact that CoCo forces the bank

to suffer from severer debt overhang resulting in later investment. The losses associated with later

investment are well offset by decreased total probability of failure and increase total value of the

total tax shields.

28

5 Sensitivity Analysis

This section examines robustness of the results obtained in section 4 by assuming alternative

to benchmark parameters’ values. In particular, I examine how the effect of CoCo on the bank’s

investment policy varies with bank’s asset volatility σ. I assume that the value of σ takes value

in the interval [0.05,0.12] which is consistent for its empirical estimates (Sundaresan and Wang

(2016)).

5.1 Investing with CoCo vs Junior Debt

In this subsection I reexamine the analysis from subsection 4.2 given alternative values of model

parameters. Tables 6, 7, and 8 summarize the bank’s optimal liability structure and optimal

investment policy for different values of asset volatility, σ, when the bank finances the exercise of

the growth option with junior debt, a PWD CoCo, and an EC CoCo, respectively.

Examining the tables reveals that CoCo’s ability to mitigate the debt overhang problem, leading

to earlier investment, is independent of the bank’s asset volatility. However, as asset volatility

becomes larger the relative difference in average time before investment under junior debt and

CoCo decreases. For example, when the bank’s asset volatility is 5% the investment under CoCo

financing is expected to happen at least 3 times earlier than under junior debt, whereas for the

bank with asset volatility of 12% only 1.3 times earlier. Therefore, CoCo’s ability to mitigate the

debt overhang problem is especially pronounced for banks with low asset volatilities.

σ D0s(Y0) DI

s(Y i) Y is Πf(Y0) ΠI(Y0) T is Fs(Y0)

0.05 0.7920 0.5000 0.0421 0.0381 0.9777 2.4059 1.46000.06 0.7361 0.5000 0.0442 0.0592 0.9642 4.6844 1.44310.07 0.6872 0.5000 0.0460 0.0893 0.9486 6.5486 1.42790.08 0.6418 0.5000 0.0479 0.1273 0.9308 8.2490 1.41430.09 0.5995 0.5000 0.0498 0.1730 0.9110 9.8100 1.40220.1 0.5603 0.5000 0.0518 0.2258 0.8896 11.2348 1.39170.11 0.5238 0.5000 0.0539 0.2850 0.8672 12.5278 1.38260.12 0.4898 0.5000 0.0560 0.3501 0.8440 13.6929 1.3748

Table 6. The bank’s optimal capital structure and investment policy as a function of the bank’svolatility when the growth option is financed by subordinated debt. D0

s(Y0) denotes the face valueof senior debt. DI

s(Y i) denotes the face value of junior debt. Y is is the optimal investment threshold.

Πf(Y0) is the unconditional probability of failure. ΠI(Y0) is the probability of investment. T is isthe expected time before investment. Fs(Y0) is the bank’s ex-ante value.

Examining the bank’s ex-ante value reveals that the bank’s preference over the loss absorption

mechanism depends on its asset volatility. The model predicts that the banks with low asst volatility

prefer a PWD CoCo over an EC CoCo, whereas banks with higher volatility have the opposite

preferences. Higher asset volatility implies riskier senior debt and CoCo, which in turn makes it

harder to satisfy the liquidity constraints. Therefor, a bank with higher asset volatility is more

liquidity constrained and this forces bank to opt for safer kind of CoCo - an EC one. It is important

29

σ D0c(Y0) DI

c(Y i) Y ic Πf(Y0) ΠI(Y0) T ic Fc(Y0)

0.05 0.7384 0.5000 0.0401 0.0008 0.9992 0.1834 1.45940.06 0.7161 0.5000 0.0402 0.0026 0.9976 0.2636 1.45340.07 0.6748 0.5000 0.0418 0.0207 0.9807 1.9903 1.43990.08 0.6117 0.5000 0.0437 0.0366 0.9671 4.1368 1.42400.09 0.5437 0.4940 0.0454 0.0491 0.9589 6.1181 1.40800.1 0.5036 0.4613 0.0475 0.0762 0.9408 8.0873 1.39280.11 0.4626 0.4291 0.0497 0.1095 0.9216 10.1861 1.37920.12 0.4275 0.3917 0.0523 0.1537 0.8987 12.3074 1.3670

Table 7. The bank’s optimal capital structure and investment policy as a function of the bank’svolatility when the growth option is financed by a PWD CoCo. D0

c(Y0) denotes the face valueof senior debt. DI

c(Y i) denotes the face value of CoCo. Y ic is the optimal investment threshold.

Πf(Y0) is the unconditional probability of failure. ΠI(Y0) is the probability of investment. T ic isthe expected time before investment. Fc(Y0) is the bank’s ex-ante value.

σ D0c(Y0) DI

c(Y i) Y ic Πf(Y0) ΠI(Y0) T ic Fc(Y0)

0.05 0.7251 0.5000 0.0406 0.0022 0.9978 0.7711 1.45500.06 0.7085 0.5000 0.0410 0.0088 0.9914 1.1939 1.44820.07 0.6862 0.5000 0.0416 0.0216 0.9800 1.7525 1.43890.08 0.6680 0.5000 0.0421 0.0425 0.9646 2.1631 1.42960.09 0.6179 0.5000 0.0445 0.0760 0.9380 4.4193 1.41620.1 0.5513 0.5000 0.0466 0.0971 0.9261 6.6301 1.40390.11 0.4771 0.5000 0.0485 0.1133 0.9220 8.7709 1.39140.12 0.4172 0.4801 0.0506 0.1415 0.9125 10.9342 1.3780

Table 8. The bank’s optimal capital structure and investment policy as a function of the bank’svolatility when the growth option is financed by an EC CoCo (λ = 2). D0

c(Y0) denotes the face valueof senior debt. DI

c(Y i) denotes the face value of CoCo. Y ic is the optimal investment threshold.

Πf(Y0) is the unconditional probability of failure. ΠI(Y0) is the probability of investment. T ic isthe expected time before investment. Fc(Y0) is the bank’s ex-ante value.

30

to highlight that without liquidity constraints the bank would always prefer a PWD CoCo since

in that case the bank will maximize the tax shields associated with it. However, without liquidity

constraints CoCo’s total coupon would require a lot of cash injection from the shareholders, which

is a very unrealistic scenario.

Interestingly, the model also predicts that the bank with asset volatility of 5% will not finance

its investment with CoCo but instead will issue junior debt, since the ex-ante value is larger under

the junior-debt financing than that under the CoCo one, despite the fact that this will result in

later investment. Although, the conditional probability of investment is higher and unconditional

probability of failure is lower under the CoCo scenario for all values of σ considered, the ex-ante

value is till higher for the junior-debt scenario thanks to much higher initial leverage for very

small value of σ. The higher initial leverage is due to later investment and relatively small cost

of resolution due to low asset volatility. Also, issuing more senior debt under CoCo scenario will

negatively affect the value of tax shields associated with the latter since CoCo converts prior to

closure, whereas tax shields from subordinated debt remain till closure.

5.2 CoCo Outstanding

In this subsection I reexamine the analysis from subsection 4.3 given alternative values of model

parameters. Tables 9, 10, and 11 summarize the bank’s optimal liability structure and optimal

investment policy for different values of asset volatility, σ, when the bank invests with junior debt,

and initially issues no CoCo, a PWD CoCo, and an EC CoCo, respectively.

Examining the tables reveals that having CoCo’s outstanding prior to investment induces later

investment regardless of the bank’s asset volatility. However, as asset volatility becomes larger

the relative difference in the average time before investment of the bank that issues CoCo prior to

investing and the bank that does not decreases. For example, when the bank’s asset volatility is

5% the investment of the bank that initially issues CoCo is expected to take place at least 2 times

later than that of the bank not issuing CoCo, whereas for the bank with asset volatility of 12%

only 1.1 times later. Therefore, the negative effect of outstanding CoCos is more pronounced for

banks with lower volatilities.

The expected time before investment is larger under EC CoCo than under PWD CoCo owning to

the presence of the equity dilution cost that comes with EC CoCo. However, the difference between

the expected times before investment under two types of CoCo decreases with asset volatility.

Examining the bank’s ex-ante value reveals that, as in the previous subsection, the bank’s

preference over the loss absorption mechanism depends on its asset volatility. The model predicts

that the banks with low asset volatility prefers a PWD CoCo over an EC CoCo, whereas banks

with higher volatility have the opposite preferences. The intuition of this results is the same as for

the case when CoCo is issued to finance investment. Higher asset volatility implies riskier senior

debt and CoCo, which in turn makes it harder to satisfy the liquidity constraints. Therefor, a bank

with higher asset volatility is more liquidity constrained and this forces bank to opt for safer kind

of CoCo - an EC one. Once again, it is important to highlight that without liquidity constraints the

31

σ D0(Y0) Y i Πf(Y0) ΠI(Y0) T i F (Y0)0.05 0.7920 0.0421 0.0381 0.9777 2.4059 1.46000.06 0.7361 0.0442 0.0592 0.9642 4.6844 1.44310.07 0.6872 0.0460 0.0893 0.9486 6.5486 1.42790.08 0.6418 0.0479 0.1273 0.9308 8.2490 1.41430.09 0.5995 0.0498 0.1730 0.9110 9.8100 1.40220.1 0.5603 0.0518 0.2258 0.8896 11.2348 1.39170.11 0.5238 0.0539 0.2850 0.8672 12.5278 1.38260.12 0.4898 0.0560 0.3501 0.8440 13.6929 1.3748

Table 9. The bank’s optimal capital structure and investment policy as a function of the bank’svolatility when the bank does not issue any CoCo at time t = 0 and the growth option is financed byjunior debt. D0(Y0) denotes the face value of senior debt. Y i is the optimal investment threshold.Πf(Y0) is the unconditional probability of failure. ΠI(Y0) is the probability of investment. T i isthe expected time before investment. F (Y0) is the bank’s ex-ante value. The bank fully financesthe exercise of the growth option by junior debt for all values of σ.

σ D0(Y0) D0c(Y0) Y i

nc Y ic Πf(Y0) ΠI(Y0) T i F (Y0)

0.05 0.7141 0.1682 0.0443 0.0419 0.0101 0.9938 4.5841 1.48730.06 0.6695 0.1649 0.0448 0.0437 0.0304 0.9905 5.3791 1.47870.07 0.6411 0.1517 0.0452 0.0455 0.0616 0.9712 6.1757 1.46300.08 0.5958 0.1473 0.0486 0.0473 0.0951 0.9550 9.1704 1.44830.09 0.5508 0.1408 0.0496 0.0492 0.1357 0.9385 10.3394 1.43440.1 0.5185 0.1294 0.0517 0.0512 0.1916 0.9151 11.9062 1.42160.11 0.4832 0.1192 0.0537 0.0532 0.2512 0.8932 13.3360 1.41000.12 0.4522 0.1086 0.0558 0.0554 0.3191 0.8691 14.5817 1.3997

Table 10. The bank’s optimal capital structure and investment policy as a function of the bank’svolatility when the bank issues a PWD CoCo at time t = 0 and the growth option is financedby junior debt. D0(Y0) denotes the face value of senior debt. D0

c(Y0) denotes the face value ofCoCo issued at time t = 0. Y i

nc is the optimal investment threshold assuming no conversion priorto investment. Y i

c is the optimal investment threshold assuming conversion prior to investment.Πf(Y0) is the unconditional probability of failure. ΠI(Y0) is the probability of investment. T i isthe expected time before investment. F (Y0) is the bank’s ex-ante value. The bank fully financesthe exercise of the growth option by junior debt for all values of σ.

32

σ D0(Y0) D0c(Y0) Y i

nc Y ic Πf(Y0) ΠI(Y0) T i F (Y0)

0.05 0.7335 0.1713 0.0464 0.0420 0.0140 0.9902 5.6589 1.48220.06 0.7181 0.1710 0.0460 0.0440 0.0470 0.9691 6.0769 1.47250.07 0.6540 0.1684 0.0504 0.0457 0.0634 0.9582 9.9460 1.45820.08 0.6033 0.1684 0.0521 0.0475 0.0938 0.9434 11.7656 1.44580.09 0.5606 0.1607 0.0536 0.0493 0.1362 0.9248 13.0857 1.43350.10 0.5190 0.1506 0.0551 0.0513 0.1859 0.9062 14.2459 1.42180.11 0.4830 0.1383 0.0566 0.0533 0.2459 0.8852 15.2015 1.41090.12 0.4500 0.1255 0.0581 0.0554 0.3133 0.8632 16.0559 1.4009

Table 11. The bank’s optimal capital structure and investment policy as a function of the bank’svolatility when the bank issues an EC CoCo (λ = 2) at time t = 0 and the growth option is financedby junior debt. D0(Y0) denotes the face value of senior debt. D0

c(Y0) denotes the face value ofCoCo issued at time t = 0. Y i

nc is the optimal investment threshold assuming no conversion priorto investment. Y i

c is the optimal investment threshold assuming conversion prior to investment.Πf(Y0) is the unconditional probability of failure. ΠI(Y0) is the probability of investment. T i isthe expected time before investment. F (Y0) is the bank’s ex-ante value. The bank fully financesthe exercise of the growth option by junior debt for all values of $\{}sigma$.

bank would always prefer a PWD CoCo since in that case the bank will maximize the tax shields

associated with it.

The ex-ante value of the bank is higher when the latter issues CoCo prior to investment regard-

less of the value of the asset volatility. Therefore, the value maximizing bank is expected to issue

CoCo prior to investment even though this leads to delayed investment.

Finally, the model predicts that it is the lower volatility banks that issue more CoCo than

their higher volatility counterparts. This model’s prediction is consistent with the empirical finding

by Goncharenko and Rauf (2016) who analyzing the sample of the EU banks from 2010-2016

confirm that among the banks issuing CoCo the ones with lower asset volatility issue more CoCo

as a percentage of the total assets.

33

A Arrow-Debreu Security Prices (incomplete)

A (Yt, Y ∗) ∶= EQt [e−r(T∗−t)] = ( YtY ∗

)β2

(27)

where T∗ ∶= inf{s ∶ Ys ≤ Y ∗} is the first time the process Yt hits the threshold Y ∗ from above and

β2 ∶= −1

σ2

⎛⎜⎝(µ − σ

2

2) +

¿ÁÁÀ(µ − σ

2

2)

2

+ 2rσ2⎞⎟⎠< 0 (28)

is the negative root of the fundamental quadratic equation.

34

B Derivation of the Valuation Equations (incomplete)

35

C The Smooth Pasting Condition for the Model from Subsec-

tion 4.2 (incomplete)

36

D The Smooth Pasting Condition for the Model from Subsec-

tion 4.3 (incomplete)

37

References

Albul, B., D.M. Jaffee, and A. Tchistyi, 2013, Contingent convertible bonds and capital structure

decisions, Working Paper .

Altman, Edward I., 1984, A further empirical investigation of the bankruptcy cost question, The

Journal of Finance 39, 1067–1089.

Avdjiev, S., P. Bolton, W. Jiang, A. Kartasheva, and B. Bogdanova, 2015, Coco bond issuance and

bank funding costs, Working Paper .

Avdjiev, S., A. Kartasheva, and B. Bogdanova, 2013, Cocos: a primer, BIS Quarterly Review .

Basel Committe, on Banking Supervision, 2010, Basel iii: A global regulatory framework for more

resilient banks and banking systems, Bank for Internaitional Settlements .

Berg, Tobias, and Christoph Kaserer, 2015, Does contingent capital induce excessive risk-taking?,

Journal of Financial Intermediation 24, 356 – 385.

Bolton, P., and F. Samama, 2012, Capital access bonds: contingent capital with an option to

convert, Economic Policy .

Bris, Arturo, Ivo Welch, and Ning Zhu, 2006, The costs of bankruptcy: Chapter 7 liquidation

versus chapter 11 reorganization, The Journal of Finance 61, 1253–1303.

Chen, N., P. Glasserman, B. Nouri, and M. Pelger, 2015, Contingent capital, tail risk, and debt-

induced collapse, Working Paper .

Flannery, M., 2011, Data-driven deposit insurance assessments, Working Paper .

Flannery, M.J., 2002, No pain, no gain? effecting market discipline via ”reverse convertible deben-

tures.”, Working Paper 352762 SSRN .

Flannery, M.J., 2005, No pain, no gain? effecting market discipline via ”reverse convertible deben-

tures.”, In Capital Adequacy Beyond Basel: Bnaking, Securities, and Insurance, ed. HS Scott,

Oxford Univ. Press .

38

Glasserman, Paul, and Behzad Nouri, 2012a, Contingent capital with a capital-ratio trigger, Man-

age. Sci. 58.

Glasserman, Paul, and Behzad Nouri, 2012b, Market-triggered changes in capital structure: Equi-

librium price dynamics, Columbia University Working Paper .

Goncharenko, Roman, and Asad Rauf, 2016, Why do not all banks issue aditional tier 1 capital?,

Working Paper .

Graham, John R., 2000, How big are the tax benefits of debt?, The Journal of Finance 55, 1901–

1941.

Hackbarth, Dirk, and David C. Mauer, 2011, Optimal priority structure, capital structure, and

investment, Review of Financial Studies .

Heckemeyer, Jost, and Ruud de Mooij, 2013, Taxation and corporate debt: are banks any different?,

Technical report.

Hennessy, Christopher A., and Toni M. Whited, 2005, Debt dynamics, The Journal of Finance 60,

1129–1165.

Hilscher, Jens, and Alon Raviv, 2014, Bank stability and market discipline: The effect of contingent

capital on risk taking and default probability, Journal of Corporate Finance 29, 542 – 560.

Himmelberg, C., and S. Tsyplakov, 2012, Incentive effects of contingent capital, Working Paper .

James, Christopher, 1991, The losses realized in bank failures, The Journal of Finance 46, 1223–

1242.

Leland, H.E., 1994, Corporate debt value, bond covenants, and optimal capital structure, The

Journal of Finance 49, 1213–1252.

Leland, H.E., and K.B. Toft, 1996, Optimal capital structure, endogenous bankruptcy, and the

term structure of credit spreads, The Journal of Finance 51, 987–1019.

McDonald, Robert L., 2013, Contingent capital with a dual price trigger, Journal of Financial

Stability 9, 230 – 241.

39

Merton, Robert C., 1974, On the pricing of corporate debt: the risk structure of interest rates*,

The Journal of Finance 29, 449–470.

Pennacchi, G., and A. Tchistyi, 2015, A reexamination of conitngent convertibles with stock price

triggers, Working Paper .

Pennacchi, G., T. Vermaelen, and C.C.P. Wolff, 2014, Contingent capital: The case of coercs,

Journal of Financial and Quantitative Analysis 49.

Pennacchi, George, 2010, A structural model of contingent bank capital, Working Paper 10-04,

FRB of Clveland .

Shibut, L., T. Critchfield, and S. Bohn, 2003, Differentiating Among Critically Undercapitalized

Banks and Thrifts, FDIC Banking Review 2, FDIC.

Squam Lake, Working Group on Financial Regulation, 2010, An expedited resolution mechanism

for distressed financial firms: regulatory hybrid securities, Squam Lake working group, Council

on Foreign Relations .

Sundaresan, S., N. Wang, and J. Yang, 2014, Dynamic investment, capital structure, and debt

overhang*, Review of Corporate Finance Studies .

Sundaresan, S., and Z. Wang, 2015, On the design of contingent capital with a market trigger,

Journal of Finance 70.

Sundaresan, S., and Z. Wang, 2016, Bank liability structure, Working Paper .

The US department of Treasury, ., 2007, Treasury conference on business taxation and global

competitiveness, Background Paper .

Zeng, Jing, 2014, Contingent capital structure, Working Paper .

40